If the normal to the point P on the ellipse intersects the major and minor axis and $G$ and $g$, then find the relation ...

Question

If the normal to the point P on the ellipse intersects the major and minor axis and $G$ and $g$, then find the relation between $CG$, $Cg$, $a$ and $b$, where C is centre

The normal at $P(x_1,y_1)$ is $$\frac{a^2x}{x_1}-\frac{b^2y}{y_1}=a^2-b^2$$

The intercepts will be $$CG=\frac{x_1(a^2-b^2)}{a^2}$$ $$Cg=\frac{y_1(a^2-b^2)}{b^2}$$

Unfortunately I am not able to eliminate the terms $x_1$ and $y_1$. How should I do it?

Answer 1

We know that $(x_1,y_1)$ must lie on the ellipse, so, we have: $$\frac{x_1^2}{a^2}+\frac{y_1^2}{b^2}=1 \leftrightarrow y_1^2=b^2\left(1-\frac{x_1^2}{a^2}\right)$$ Substituing, we obtain: $$Cg^2=\frac{b^2\left(1-\frac{x_1^2}{a^2}\right)(a^2-b^2)^2}{b^4}$$ While $CG^2$ is: $$CG^2=\frac{x_1^2(a^2-b^2)^2}{a^4}\leftrightarrow x_1^2=\frac{CG^2\cdot a^4}{(a^2-b^2)^2} $$ Substituing again, you arrive at: $$Cg^2=\frac{b^2\left(1-\frac{CG^2\cdot a^4}{a^2(a^2-b^2)^2}\right)(a^2-b^2)^2}{b^4} \rightarrow Cg^2=\frac{(a^2-b^2)^2-CG^2\cdot a^2}{b^2}$$ From here, it's very simple. In fact: $$Cg=\sqrt{Cg^2}=\sqrt{\frac{(a^2-b^2)^2-CG^2\cdot a^2}{b^2}}=\frac{\sqrt{(a^2-b^2)^2-CG^2\cdot a^2}}{b}$$

Note that, as @Blue suggested, you can rewrite the relation for $Cg^2$ as: $$\frac{Cg^2}{a^2} + \frac{CG^2}{b^2} =\frac{ (a^2-b^2)^2}{a^2b^2}$$ This follows from here: $$Cg^2=\frac{(a^2-b^2)^2-CG^2\cdot a^2}{b^2} \leftrightarrow Cg^2b^2-CGa^2=(a^2-b^2)^2 \leftrightarrow \frac{Cg^2}{a^2} + \frac{CG^2}{b^2} =\frac{ (a^2-b^2)^2}{a^2b^2}$$

Answer 2

WLOG any point on the ellipse $P(a\cos t,b\sin t)$

Using https://www.askiitians.com/iit-jee-coordinate-geometry/tangent-and-normal.aspx#how-do-you-find-the-equation-of-the-normal-to-an-ellipse ,

the equation of the normal $$ax\sec t-by\csc t=a^2-b^2$$

$G: \dfrac{(a^2-b^2)\cos t}a,0$

$g: 0, -\dfrac{(a^2-b^2)\sin t}b$

$$a\cdot CG=|(a^2-b^2)\cos t|,|\cos t|=?$$

Similarly

$$b\cdot CG=|(a^2-b^2\sin t|,|\sin t|=?$$

Eliminate $t$ using $\cos^2t+\sin^2t=1$$

Answer 3

In the same vein as Matteo’s answer but with, I think, much less work: From your calculations we have $$x_1 = {a^2\over a^2-b^2}{CG} \\ y_1 = {b^2\over a^2-b^2}{Cg}.$$ The point $(x_1,y_1)$ lies on the ellipse, therefore $${x_1^2\over a^2} + {y_1^2\over b^2} = \frac1{a^2}\left({a^2\over a^2-b^2}{CG}\right)^2 + \frac1{b^2}\left({b^2\over a^2-b^2}{Cg}\right)^2 = 1$$ and after some straightforward rearrangement, we end up with $${{Cg}^2\over a^2}+{{CG}^2\over b^2} = {(a^2-b^2)^2\over a^2b^2}.$$